First responders who became ill after working at Ground Zero deserve every penny of compensation they get. But does that mean anyone claiming to be such a person should automatically be believed?

That’s the question behind a new controversy at the 9/11 Museum over the history of that attack and its aftermath. And it’s the latest test of whether that history will be skewed by emotions.

According to reports, some first-responders and their families object to a descriptive panel at the museum that says the Zadroga 9/11 Health and Compensation Act was meant to ensure medical treatment and financial compensation “for those with health conditions claimed to be related to the World Trade Center Disaster.”

The objections focus on the word “claimed.”

Bad enough, critics say, that there’s no explicit listing of victims of 9/11-related illnesses; the word “claim” casts doubt on the true source of their illnesses.

Yet doubt is sometimes warranted. For one thing, in a world where fraud is hardly unheard of, not every applicant should automatically be assumed to be telling the truth.

For another, assumptions about the source of an illness can later prove wrong.

In one notable case, for example, Police Officer Cesar Borja was said to have “rushed” to the World Trade Center site after the attacks and worked “16-hour shifts.” After his death of lung disease, he became the poster-child for 9/11 ailments.

Later, it came out that Borja had only first gone to the WTC site in December of 2001, nearly three months after 9/11. And he primarily directed traffic blocks away.

Even the death of James Zadroga, for whom the law was named, was debated. A New Jersey medical examiner said he suffered 9/11-related lung disease.

But New York City’s ME cited ground-up prescription drugs injected into his veins.

Here’s the key point: No one wants to deny a true hero the compensation — and recognition — he or she deserves. But it doesn’t help true heroes to say that anyone claiming to be one truly is one.